Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces

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September 2020, 40(9): 5289-5323. doi: 10.3934/dcds.2020228

Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces

Hongjie Dong ^, and Kunrui Wang

Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA

^* Corresponding author: Hongjie Dong

Received October 2019 Revised April 2020 Published June 2020

Fund Project: H. Dong and K. Wang were partially supported by the NSF under agreement DMS-1600593

We study regularity criteria for the $ d $-dimensional incompressible Navier-Stokes equations. We prove if $ u\in L_{\infty}^tL_d^x((0,T)\times{\mathbb{R}}^d_+) $ is a Leray-Hopf weak solution vanishing on the boundary, then $ u $ is regular up to the boundary in $ (0,T)\times {\mathbb{R}}^d_+ $. Furthermore, with a stronger uniform local condition on the pressure $ p $, we prove $ u $ is unique and tends to zero as $ t\rightarrow \infty $ if $ T = \infty $. This generalizes a result by Escauriaza, Seregin, and Šverák [14] to higher dimensions and domains with boundary. We also study the local problem in half unit cylinder $ Q^+ $ and prove that if $ u\in L^t_{\infty}L^x_d(Q^+) $ and $ p\in L_{2-1/d}(Q^+) $, then $ u $ is Hölder continuous in the closure of the set $ Q^+(1/4) $.

Keywords: Incompressible Navier-Stokes equations, Leray-Hopf weak solutions, suitable weak solutions, regularity criteria, critical Lebesgue spaces.

Mathematics Subject Classification: Primary: 35Q30, 35B65; Secondary: 76D05.

Citation: Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228

References:

[1]	D. Albritton and T. Barker, Global weak Besov solutions of the Navier-Stokes equations and applications, e-prints, (2018).
[2]	D. Albritton, Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces, Anal. PDE, 11 (2018), 1415-1456. doi: 10.2140/apde.2018.11.1415.
[3]	T. Barker and G. Seregin, A necessary condition of potential blowup for the Navier-Stokes system in half-space, Math. Ann., 369 (2017), 1327-1352. doi: 10.1007/s00208-016-1488-9.
[4]	L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604.
[5]	A. Cheskidov and R. Shvydkoy, The regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\infty,\infty}$, Arch. Ration. Mech. Anal., 195 (2010), 159-169. doi: 10.1007/s00205-009-0265-2.
[6]	F. Chiarenza, M. Frasca and P. Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat., 40 (1991), 149-168.
[7]	H. J. Dong and D. P. Du, Partial regularity of solutions to the four-dimensional Navier-Stokes equations at the first blow-up time, Comm. Math. Phys., 273 (2007), 785-801. doi: 10.1007/s00220-007-0259-6.
[8]	H. J. Dong and D. P. Du, The Navier-Stokes equations in the critical Lebesgue space, Comm. Math. Phys., 292 (2009), 811-827. doi: 10.1007/s00220-009-0852-y.
[9]	H. J. Dong and X. M. Gu, Partial regularity of solutions to the four-dimensional Navier-Stokes equations, Dyn. Partial Differ. Equ., 11 (2014), 53-69. doi: 10.4310/DPDE.2014.v11.n1.a3.
[10]	H. J. Dong and X. M. Gu, Boundary partial regularity for the high dimensional Navier-Stokes equations, J. Funct. Anal., 267 (2014), 2606-2637. doi: 10.1016/j.jfa.2014.08.001.
[11]	H. J. Dong and D. Li, Optimal local smoothing and analyticity rate estimates for the generalized Navier-Stokes equations, Commun. Math. Sci., 7 (2009), 67-80. doi: 10.4310/CMS.2009.v7.n1.a3.
[12]	H. J. Dong and R. M. Strain, On partial regularity of steady-state solutions to the 6D Navier-Stokes equations, Indiana Univ. Math. J., 61 (2012), 2211-2229. doi: 10.1512/iumj.2012.61.4765.
[13]	H. J. Dong and K. R. Wang, Boundary $\varepsilon$-regularity criteria for the 3D Navier-Stokes equations, SIAM J. Math. Anal., 52 (2020), 1290-1309. doi: 10.1137/18M1234722.
[14]	L. Escauriaza, G. A. Sëregin and V. Sverak, Sëregin-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44. doi: 10.1070/RM2003v058n02ABEH000609.
[15]	L. Escauriaza, G. Seregin and V. Šverák, On backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169 (2003), 147-157. doi: 10.1007/s00205-003-0263-8.
[16]	I. Gallagher, G. S. Koch and F. Planchon, Blow-up of critical Besov norms at a potential Navier-Stokes singularity, Comm. Math. Phys., 343 (2016), 39-82. doi: 10.1007/s00220-016-2593-z.
[17]	M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ, 1983.
[18]	Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.
[19]	Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875.
[20]	Y. Giga and O. Sawada, On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem, Nonlinear Analysis and Applications, Kluwer Acad. Publ., Dordrecht, 1,2 (2003), 549-562.
[21]	C. Guevara and N. C. Phuc, Local energy bounds and $\epsilon$-regularity criteria for the 3D Navier-Stokes system, Calc. Var. Partial Differential Equations, 56 (2017), Art. 68, 16 pp. doi: 10.1007/s00526-017-1151-7.
[22]	E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231. doi: 10.1002/mana.3210040121.
[23]	C. Kahane, On the spatial analyticity of solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 33 (1969), 386-405. doi: 10.1007/BF00247697.
[24]	T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in R^m, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.
[25]	C. E. Kenig and G. S. Koch, An alternative approach to regularity for the Navier-Stokes equations in critical spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 159-187. doi: 10.1016/j.anihpc.2010.10.004.
[26]	H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.
[27]	O. A. Ladyženskaja, Uniqueness and smoothness of generalized solutions of Navier-Stokes equations, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 5 (1967), 169-185.
[28]	O. A. Ladyzhenskaya and G. A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 356-387. doi: 10.1007/s000210050015.
[29]	J. Leray, Étude de diverses équations intérales non linéaires et de quelques problemes que pose l’hydrodynamique, NUMDAM, (1933), 82 pp.
[30]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[31]	F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.
[32]	P. Maremonti and V. A. Solonnikov, On estimates for the solutions of the nonstationary Stokes problem in S. L. Sobolev anisotropic spaces with a mixed norm, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 222 (1995), no. Issled. po Lineĭn. Oper. i Teor. Funktsiĭ. 23,124-150,309. doi: 10.1007/BF02355828.
[33]	K. Masuda, On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation, Proc. Japan Acad., 43 (1967), 827-832. doi: 10.3792/pja/1195521421.
[34]	A. S. Mikhailov and T. N. Shilkin, $L_3,\infty$-solutions to the 3D-Navier-Stokes system in the domain with a curved boundary, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 336 (2006), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 37,133–152,276. doi: 10.1007/s10958-007-0176-4.
[35]	N. C. Phuc, The Navier-Stokes equations in nonendpoint borderline Lorentz spaces, J. Math. Fluid Mech., 17 (2015), 741-760. doi: 10.1007/s00021-015-0229-2.
[36]	G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl. (4), 48 (1959), 173-182. doi: 10.1007/BF02410664.
[37]	V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552. doi: 10.2140/pjm.1976.66.535.
[38]	V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys., 55 (1977), 97-112. doi: 10.1007/BF01626512.
[39]	V. Scheffer, The Navier-Stokes equations on a bounded domain, Comm. Math. Phys., 73 (1980), 1-42. doi: 10.1007/BF01942692.
[40]	G. A. Seregin, Some estimates near the boundary for solutions to the non-stationary linearized Navier-Stokes equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 271 (2000), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 31,204–223,317. doi: 10.1023/A:1023330105200.
[41]	G. A. Seregin, Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary, J. Math. Fluid Mech., 4 (2002), 1-29. doi: 10.1007/s00021-002-8533-z.
[42]	G. Seregin, On smoothness of $L_3,\infty$-solutions to the Navier-Stokes equations up to boundary, Math. Ann., 332 (2005), 219-238. doi: 10.1007/s00208-004-0625-z.
[43]	G. A. Seregin, A note on local boundary regularity for the Stokes system, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 370 (2009), Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 40, 151–159, 221–222. doi: 10.1007/s10958-010-9847-7.
[44]	G. A. Seregin, T. N. Shilkin and V. A. Solonnikov, Boundary partial regularity for the Navier-Stokes equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34], 158–190,228. doi: 10.1007/s10958-005-0502-7.
[45]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195. doi: 10.1007/BF00253344.
[46]	J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems, Univ. of Wisconsin Press, Madison, Wis., (1963), 69–98.
[47]	M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. doi: 10.1002/cpa.3160410404.
[48]	M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations, 17 (1992), 1407-1456. doi: 10.1080/03605309208820892.
[49]	A. F. Vasseur, A new proof of partial regularity of solutions to Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 753-785. doi: 10.1007/s00030-007-6001-4.
[50]	W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Aspects of Mathematics, E8, Friedr. Vieweg & Sohn, Braunschweig, 1985. doi: 10.1007/978-3-663-13911-9.
[51]	W. D. Wang and Z. F. Zhang, Blow-up of critical norms for the 3-D Navier-Stokes equations, Sci. China Math., 60 (2017), 637-650. doi: 10.1007/s11425-016-0344-5.
[52]	F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Rational Mech. Anal., 74 (1980), 219-230. doi: 10.1007/BF00280539.

show all references

References:

[1]	D. Albritton and T. Barker, Global weak Besov solutions of the Navier-Stokes equations and applications, e-prints, (2018).
[2]	D. Albritton, Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces, Anal. PDE, 11 (2018), 1415-1456. doi: 10.2140/apde.2018.11.1415.
[3]	T. Barker and G. Seregin, A necessary condition of potential blowup for the Navier-Stokes system in half-space, Math. Ann., 369 (2017), 1327-1352. doi: 10.1007/s00208-016-1488-9.
[4]	L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604.
[5]	A. Cheskidov and R. Shvydkoy, The regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\infty,\infty}$, Arch. Ration. Mech. Anal., 195 (2010), 159-169. doi: 10.1007/s00205-009-0265-2.
[6]	F. Chiarenza, M. Frasca and P. Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat., 40 (1991), 149-168.
[7]	H. J. Dong and D. P. Du, Partial regularity of solutions to the four-dimensional Navier-Stokes equations at the first blow-up time, Comm. Math. Phys., 273 (2007), 785-801. doi: 10.1007/s00220-007-0259-6.
[8]	H. J. Dong and D. P. Du, The Navier-Stokes equations in the critical Lebesgue space, Comm. Math. Phys., 292 (2009), 811-827. doi: 10.1007/s00220-009-0852-y.
[9]	H. J. Dong and X. M. Gu, Partial regularity of solutions to the four-dimensional Navier-Stokes equations, Dyn. Partial Differ. Equ., 11 (2014), 53-69. doi: 10.4310/DPDE.2014.v11.n1.a3.
[10]	H. J. Dong and X. M. Gu, Boundary partial regularity for the high dimensional Navier-Stokes equations, J. Funct. Anal., 267 (2014), 2606-2637. doi: 10.1016/j.jfa.2014.08.001.
[11]	H. J. Dong and D. Li, Optimal local smoothing and analyticity rate estimates for the generalized Navier-Stokes equations, Commun. Math. Sci., 7 (2009), 67-80. doi: 10.4310/CMS.2009.v7.n1.a3.
[12]	H. J. Dong and R. M. Strain, On partial regularity of steady-state solutions to the 6D Navier-Stokes equations, Indiana Univ. Math. J., 61 (2012), 2211-2229. doi: 10.1512/iumj.2012.61.4765.
[13]	H. J. Dong and K. R. Wang, Boundary $\varepsilon$-regularity criteria for the 3D Navier-Stokes equations, SIAM J. Math. Anal., 52 (2020), 1290-1309. doi: 10.1137/18M1234722.
[14]	L. Escauriaza, G. A. Sëregin and V. Sverak, Sëregin-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44. doi: 10.1070/RM2003v058n02ABEH000609.
[15]	L. Escauriaza, G. Seregin and V. Šverák, On backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169 (2003), 147-157. doi: 10.1007/s00205-003-0263-8.
[16]	I. Gallagher, G. S. Koch and F. Planchon, Blow-up of critical Besov norms at a potential Navier-Stokes singularity, Comm. Math. Phys., 343 (2016), 39-82. doi: 10.1007/s00220-016-2593-z.
[17]	M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ, 1983.
[18]	Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.
[19]	Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875.
[20]	Y. Giga and O. Sawada, On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem, Nonlinear Analysis and Applications, Kluwer Acad. Publ., Dordrecht, 1,2 (2003), 549-562.
[21]	C. Guevara and N. C. Phuc, Local energy bounds and $\epsilon$-regularity criteria for the 3D Navier-Stokes system, Calc. Var. Partial Differential Equations, 56 (2017), Art. 68, 16 pp. doi: 10.1007/s00526-017-1151-7.
[22]	E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231. doi: 10.1002/mana.3210040121.
[23]	C. Kahane, On the spatial analyticity of solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 33 (1969), 386-405. doi: 10.1007/BF00247697.
[24]	T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in R^m, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.
[25]	C. E. Kenig and G. S. Koch, An alternative approach to regularity for the Navier-Stokes equations in critical spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 159-187. doi: 10.1016/j.anihpc.2010.10.004.
[26]	H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.
[27]	O. A. Ladyženskaja, Uniqueness and smoothness of generalized solutions of Navier-Stokes equations, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 5 (1967), 169-185.
[28]	O. A. Ladyzhenskaya and G. A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 356-387. doi: 10.1007/s000210050015.
[29]	J. Leray, Étude de diverses équations intérales non linéaires et de quelques problemes que pose l’hydrodynamique, NUMDAM, (1933), 82 pp.
[30]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[31]	F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.
[32]	P. Maremonti and V. A. Solonnikov, On estimates for the solutions of the nonstationary Stokes problem in S. L. Sobolev anisotropic spaces with a mixed norm, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 222 (1995), no. Issled. po Lineĭn. Oper. i Teor. Funktsiĭ. 23,124-150,309. doi: 10.1007/BF02355828.
[33]	K. Masuda, On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation, Proc. Japan Acad., 43 (1967), 827-832. doi: 10.3792/pja/1195521421.
[34]	A. S. Mikhailov and T. N. Shilkin, $L_3,\infty$-solutions to the 3D-Navier-Stokes system in the domain with a curved boundary, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 336 (2006), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 37,133–152,276. doi: 10.1007/s10958-007-0176-4.
[35]	N. C. Phuc, The Navier-Stokes equations in nonendpoint borderline Lorentz spaces, J. Math. Fluid Mech., 17 (2015), 741-760. doi: 10.1007/s00021-015-0229-2.
[36]	G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl. (4), 48 (1959), 173-182. doi: 10.1007/BF02410664.
[37]	V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552. doi: 10.2140/pjm.1976.66.535.
[38]	V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys., 55 (1977), 97-112. doi: 10.1007/BF01626512.
[39]	V. Scheffer, The Navier-Stokes equations on a bounded domain, Comm. Math. Phys., 73 (1980), 1-42. doi: 10.1007/BF01942692.
[40]	G. A. Seregin, Some estimates near the boundary for solutions to the non-stationary linearized Navier-Stokes equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 271 (2000), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 31,204–223,317. doi: 10.1023/A:1023330105200.
[41]	G. A. Seregin, Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary, J. Math. Fluid Mech., 4 (2002), 1-29. doi: 10.1007/s00021-002-8533-z.
[42]	G. Seregin, On smoothness of $L_3,\infty$-solutions to the Navier-Stokes equations up to boundary, Math. Ann., 332 (2005), 219-238. doi: 10.1007/s00208-004-0625-z.
[43]	G. A. Seregin, A note on local boundary regularity for the Stokes system, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 370 (2009), Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 40, 151–159, 221–222. doi: 10.1007/s10958-010-9847-7.
[44]	G. A. Seregin, T. N. Shilkin and V. A. Solonnikov, Boundary partial regularity for the Navier-Stokes equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34], 158–190,228. doi: 10.1007/s10958-005-0502-7.
[45]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195. doi: 10.1007/BF00253344.
[46]	J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems, Univ. of Wisconsin Press, Madison, Wis., (1963), 69–98.
[47]	M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. doi: 10.1002/cpa.3160410404.
[48]	M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations, 17 (1992), 1407-1456. doi: 10.1080/03605309208820892.
[49]	A. F. Vasseur, A new proof of partial regularity of solutions to Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 753-785. doi: 10.1007/s00030-007-6001-4.
[50]	W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Aspects of Mathematics, E8, Friedr. Vieweg & Sohn, Braunschweig, 1985. doi: 10.1007/978-3-663-13911-9.
[51]	W. D. Wang and Z. F. Zhang, Blow-up of critical norms for the 3-D Navier-Stokes equations, Sci. China Math., 60 (2017), 637-650. doi: 10.1007/s11425-016-0344-5.
[52]	F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Rational Mech. Anal., 74 (1980), 219-230. doi: 10.1007/BF00280539.

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